Finite Group Databasebeta

← #35331007

Subgroups — #35331007

Order 21248.

1502 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581782C211yesmaximal, characteristic
2C2×#19127570C211yesmaximal, characteristic
3C2×#19127546C211yesmaximal, characteristic
4#24905717C211yesmaximal, characteristic
5#24905716C211yesmaximal, characteristic
6#24905715C212yesmaximal
7#24905714C212yesmaximal
8#24905713C211yesmaximal, characteristic
9Dic83⋊#95C212yesmaximal
10Dic83⋊#95C212yesmaximal
11C83×#1441C211yesmaximal, characteristic
12C2×#14581782—24no—
13C2×#14581782—24no—
14C2×#14581782C418yescomplemented
15C22×C4⋊Dic83C2211yescharacteristic
16C22×C4⋊Dic83C2211yescharacteristic
17C22×Dic83⋊C4C2212yes—
18C22×Dic83⋊C4C2211yescharacteristic
19C22×Dic83⋊C4C2211yescharacteristic
20C22×Dic83⋊C4C2211yescharacteristic
21C22×Dic83⋊C4C2211yescharacteristic
22C22×Dic83⋊C4C2212yes—
23C22×C4×Dic83C2212yes—
24C22×C4×Dic83C2211yescharacteristic
25#19127570—24no—
26#19127546—24no—
27C2×C83⋊#95C2211yescharacteristic
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2212yes—
32C2×C83⋊#95C2212yes—
33C2×C83⋊#95C2212yes—
34C2×C83⋊#95C2212yes—
35C22×C4⋊C4×C83C2211yescharacteristic
36C22×C4⋊C4×C83C2211yescharacteristic
37C2×#95×C83C2211yescharacteristic
38C2×#95×C83C2212yes—
39C2×#95×C83C2212yes—
40C23×Dic83C2312yes—
41C23×Dic83C2311yescharacteristic
42C23×Dic83C2311yescharacteristic
43C23×Dic83C2312yes—
44C23×Dic83C2311yescharacteristic
45C23×Dic83C2311yescharacteristic
46#14581782—232no—
47C2×C4⋊Dic83—22no—
48C2×C4⋊Dic83—22no—
49C2×C4⋊Dic83—22no—
50C2×C4⋊Dic83—42no—
51C2×C4⋊Dic83C2×C414yes—
52C2×Dic83⋊C4—22no—
53C2×Dic83⋊C4—22no—
54C2×Dic83⋊C4—22no—
55C2×Dic83⋊C4—22no—
56C2×Dic83⋊C4—22no—
57C2×Dic83⋊C4—22no—
58C2×Dic83⋊C4—24no—
59C2×Dic83⋊C4—24no—
60C2×Dic83⋊C4—24no—
61C2×Dic83⋊C4—24no—
62C2×Dic83⋊C4—22no—
63C2×Dic83⋊C4—22no—
64C2×Dic83⋊C4—42no—
65C2×Dic83⋊C4—42no—
66C2×Dic83⋊C4C2×C414yes—
67C2×Dic83⋊C4C2×C414yes—
68C2×Dic83⋊C4C2×C414yes—
69C2×Dic83⋊C4C2×C414yes—
70C2×C4×Dic83—24no—
71C2×C4×Dic83—24no—
72C2×C4×Dic83C2×C414yes—
73C2×C4×Dic83D414yes—
74C2×C4×Dic83D414yes—
75C2×C4×Dic83Q818yes—
76C83⋊#95—24no—
77C83⋊#95—22no—
78C83⋊#95—42no—
79C83⋊#95—42no—
80C83⋊#95—42no—
81C83⋊#95—42no—
82C83⋊#95—42no—
83C83⋊#95—42no—
84C23×C4×C83C2312yes—
85C23×C4×C83C2311yescharacteristic
86C23×C4×C83C2311yescharacteristic
87C23×C4×C83C2312yes—
88C23×C4×C83C2311yescharacteristic
89C2×C4⋊C4×C83—22no—
90C2×C4⋊C4×C83—22no—
91C2×C4⋊C4×C83—22no—
92C2×C4⋊C4×C83—42no—
93C2×C4⋊C4×C83C2×C414yes—
94#95×C83—22no—
95#95×C83—42no—
96#95×C83—42no—
97C22×Dic83—21no—
98C22×Dic83—21no—
99C22×Dic83—21no—
100C22×Dic83—21no—
101C22×Dic83—22no—
102C22×Dic83—21no—
103C22×Dic83—21no—
104C22×Dic83—21no—
105C22×Dic83—21no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—21no—
109C22×Dic83—22no—
110C22×Dic83—21no—
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—21no—
114C22×Dic83—22no—
115C22×Dic83—21no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—22no—
121C22×Dic83—22no—
122C22×Dic83—22no—
123C22×Dic83—22no—
124C22×Dic83C22×C412yes—
125C22×Dic83C22×C412yes—
126C22×Dic83C22×C412yes—
127C22×Dic83C22×C412yes—
128C22×Dic83C2×D412yes—
129C22×Dic83C2×D412yes—
130C22×Dic83C2×D412yes—
131C22×Dic83C2×D412yes—
132C22×Dic83C2×Q814yes—
133C22×Dic83C2×Q814yes—
134C22×Dic83#4114yes—
135C4⋊Dic83—28no—
136C4⋊Dic83—44no—
137Dic83⋊C4—28no—
138Dic83⋊C4—28no—
139Dic83⋊C4—28no—
140Dic83⋊C4—28no—
141Dic83⋊C4—48no—
142Dic83⋊C4—48no—
143C4×Dic83—216no—
144C4×Dic83C22⋊C4116yes—
145C24×C83C2411yescharacteristic
146C22×C4×C83—22no—
147C22×C4×C83—22no—
148C22×C4×C83—22no—
149C22×C4×C83—21no—
150C22×C4×C83—21no—
151C22×C4×C83—21no—
152C22×C4×C83—21no—
153C22×C4×C83—22no—
154C22×C4×C83—21no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—22no—
158C22×C4×C83—21no—
159C22×C4×C83—21no—
160C22×C4×C83—22no—
161C22×C4×C83—22no—
162C22×C4×C83—22no—
163C22×C4×C83—21no—
164C22×C4×C83—22no—
165C22×C4×C83—22no—
166C22×C4×C83—22no—
167C22×C4×C83—22no—
168C22×C4×C83—22no—
169C22×C4×C83—22no—
170C22×C4×C83—22no—
171C22×C4×C83C22×C412yes—
172C22×C4×C83C22×C412yes—
173C22×C4×C83C22×C412yes—
174C22×C4×C83C2×D412yes—
175C22×C4×C83C2×D412yes—
176C22×C4×C83C2×Q814yes—
177C4⋊C4×C83—28no—
178C4⋊C4×C83—44no—
179C23×C83C23×C411yescharacteristic
180C23×C83C22×D411yescharacteristic
181C23×C83C22×D411yescharacteristic
182C23×C83C22×Q812yes—
183C23×C83C2×#4112yes—
184C23×C83H(5,𝔽2)12yes—
185C23×C83H(5,𝔽2)11yescharacteristic
186C23×C83H(5,𝔽2)11yescharacteristic
187C23×C832−1+411yescharacteristic
188C23×C832−1+412yes—
189C23×C832−1+411yescharacteristic
190C2×C4×C83—24no—
191C2×C4×C83—22no—
192C2×C4×C83—22no—
193C2×C4×C83—24no—
194C2×C4×C83—22no—
195C2×C4×C83—22no—
196C2×C4×C83—24no—
197C2×C4×C83—22no—
198C2×C4×C83—22no—
199C2×C4×C83—24no—
200C2×C4×C83—24no—
201C2×C4×C83—24no—
202C2×C4×C83—41no—
203C2×C4×C83—41no—
204C2×C4×C83—41no—
205C2×C4×C83—41no—
206C2×C4×C83—42no—
207C2×C4×C83—42no—
208C2×C4×C83—41no—
209C2×C4×C83—42no—
210C2×C4×C83—41no—
211C2×C4×C83—42no—
212C2×C4×C83—42no—
213C2×C4×C83—42no—
214C2×C4×C83—42no—
215C2×C4×C83—42no—
216C2×C4×C83—42no—
217C2×C4×C83—42no—
218C2×C4×C83—42no—
219C2×C4×C83—42no—
220C2×C4×C83C2×C22⋊C414yes—
221C2×Dic83—22no—
222C2×Dic83—22no—
223C2×Dic83—22no—
224C2×Dic83—22no—
225C2×Dic83—22no—
226C2×Dic83—22no—
227C2×Dic83—22no—
228C2×Dic83—22no—
229C2×Dic83—24no—
230C2×Dic83—22no—
231C2×Dic83—22no—
232C2×Dic83—24no—
233C2×Dic83—24no—
234C2×Dic83—24no—
235C2×Dic83—41no—
236C2×Dic83—41no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—42no—
240C2×Dic83—42no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—42no—
244C2×Dic83—42no—
245C2×Dic83—42no—
246C2×Dic83—42no—
247C2×Dic83—42no—
248C2×Dic83—42no—
249C2×Dic83—42no—
250C2×Dic83—42no—
251C2×Dic83—42no—
252C2×Dic83—42no—
253C2×Dic83C2×C22⋊C414yes—
254C2×Dic83C2×C22⋊C414yes—
255C2×Dic83C4×Q818yes—
256C2×Dic83C22⋊Q818yes—
257C2×Dic83#12414yes—
258C2×Dic83C4⋊Q814yes—
259C22×C83C22×C22⋊C411yescharacteristic
260C22×C83C2×C4×Q812yes—
261C22×C83#52012yes—
262C22×C83#52011yescharacteristic
263C22×C83#52011yescharacteristic
264C22×C83C2×C22⋊Q812yes—
265C22×C83C2×#12411yescharacteristic
266C22×C83C2×C4⋊Q811yescharacteristic
267C22×C83#53511yescharacteristic
268C22×C83#53511yescharacteristic
269C22×C83#53611yescharacteristic
270C22×C83#53611yescharacteristic
271C22×C83#53712yes—
272C22×C83#53711yescharacteristic
273C22×C83#53711yescharacteristic
274C22×C83#54012yes—
275C22×C83#54012yes—
276C22×C83#54112yes—
277C22×C83#54112yes—
278C22×C83#54412yes—
279C22×C83#54412yes—
280C22×C83#54412yes—
281C22×C83#54412yes—
282C4×C83—24no—
283C4×C83—42no—
284C4×C83—42no—
285C4×C83—44no—
286C4×C83—82no—
287Dic83—24no—
288Dic83—42no—
289Dic83—42no—
290Dic83—44no—
291Dic83—82no—
292Dic83#39118yes—
293#14283—831nomaximal, Sylow
294C2×C83C2×#39111yescharacteristic
295C2×C83#231111yescharacteristic
296C2×C83#231111yescharacteristic
297C2×C83#233012yes—
298C2×C83#233012yes—
299C2×C83#262912yes—
300C2×C83#262912yes—
301C2×C83#264111yescharacteristic
302C2×C83#264211yescharacteristic
303C2×C83#266011yescharacteristic
304C2×C83#266011yescharacteristic
305C22×#125—831no—
306C2×#384—831no—
307C2×#376—831no—
308#1441—832no—
309#1441—832no—
310#1441—831no—
311#1441—831no—
312#1441—831no—
313#1441D8311yescharacteristic
314C4⋊#95—832no—
315C4⋊#95—832no—
316C83#1428311yescharacteristic, complemented, Sylow
317C2×#125—838no—
318C2×#125—1664no—
319C2×#125—1664no—
320C22×C4⋊C4—832no—
321C22×C4⋊C4—831no—
322C22×C4⋊C4—831no—
323C22×C4⋊C4—831no—
324C22×C4⋊C4—831no—
325C22×C4⋊C4—831no—
326C22×C4⋊C4—831no—
327C22×C4⋊C4—832no—
328C22×C4⋊C4C2×D8311yescharacteristic
329C22×C4⋊C4C2×D8311yescharacteristic
330C22×C42—831no—
331C22×C42—832no—
332#384—1664no—
333#376—1664no—
334C2×#95—832no—
335C2×#95—832no—
336C2×#95—832no—
337C2×#95—832no—
338C2×#95—832no—
339C2×#95—832no—
340C2×#95—832no—
341C2×#95—831no—
342C2×#95C2×D8311yescharacteristic
343C2×#95C2×D8312yes—
344C2×#95C2×D8312yes—
345C23×C4—831no—
346C23×C4—831no—
347C23×C4—831no—
348C23×C4—831no—
349C23×C4—832no—
350C23×C4—832no—
351C23×C4C22×D8312yes—
352C23×C4C22×D8311yescharacteristic
353C23×C4C22×D8311yescharacteristic
354C23×C4C22×D8312yes—
355C23×C4C22×D8311yescharacteristic
356#125—16632no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—42no—
361C2×C4⋊C4—3322no—
362C2×C4⋊C4—3322no—
363C2×C4⋊C4—3322no—
364C2×C4⋊C4—834no—
365C2×C4⋊C4—834no—
366C2×C4⋊C4—834no—
367C2×C4⋊C4—834no—
368C2×C4⋊C4—834no—
369C2×C4⋊C4—1664no—
370C2×C4⋊C4—1664no—
371C2×C4⋊C4—1664no—
372C2×C4⋊C4—1664no—
373C2×C4⋊C4—1662no—
374C2×C4⋊C4—1662no—
375C2×C4⋊C4—1662no—
376C2×C4⋊C4—1662no—
377C2×C4⋊C4—1662no—
378C2×C4⋊C4—1662no—
379C2×C4⋊C4—1662no—
380C2×C4⋊C4—1662no—
381C2×C4⋊C4—1662no—
382C2×C4⋊C4—1662no—
383C2×C4⋊C4—1662no—
384C2×C4⋊C4C4×D8314yes—
385C2×C42—834no—
386C2×C42—834no—
387C2×C42—834no—
388C2×C42—838no—
389C2×C42—1664no—
390C2×C42—1664no—
391#95—22no—
392#95—42no—
393#95—42no—
394#95—3322no—
395#95—3322no—
396#95—3322no—
397#95—3322no—
398#95—3322no—
399#95—3322no—
400#95—1664no—
401#95—1662no—
402C24C23×D8311yescharacteristic
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—832no—
429C22×C4—832no—
430C22×C4—832no—
431C22×C4—834no—
432C22×C4—834no—
433C22×C4—832no—
434C22×C4—832no—
435C22×C4—832no—
436C22×C4—832no—
437C22×C4—832no—
438C22×C4—834no—
439C22×C4—1662no—
440C22×C4—1662no—
441C22×C4—1662no—
442C22×C4—1662no—
443C22×C4—1662no—
444C22×C4—1662no—
445C22×C4—1662no—
446C22×C4—1662no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4—1661no—
450C22×C4—1661no—
451C22×C4—1662no—
452C22×C4—1662no—
453C22×C4—1661no—
454C22×C4—1662no—
455C22×C4—1661no—
456C22×C4—1662no—
457C22×C4—1662no—
458C22×C4—1661no—
459C22×C4—1662no—
460C22×C4—1661no—
461C22×C4—1662no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466C22×C4C2×C4×D8312yes—
467C22×C4C2×C4×D8312yes—
468C22×C4C2×C4×D8312yes—
469C22×C4D4×D8312yes—
470C22×C4D4×D8312yes—
471C22×C4Q8×D8314yes—
472C4⋊C4—28no—
473C4⋊C4—44no—
474C4⋊C4—3324no—
475C4⋊C4—3328no—
476C4⋊C4—3328no—
477C4⋊C4—1668no—
478C4⋊C4—1668no—
479C4⋊C4—1668no—
480C4⋊C4—1668no—
481C4⋊C4—1668no—
482C42—8316no—
483C42—16616no—
484C23C22×C4×D8311yescharacteristic
485C23C2×D4×D8311yescharacteristic
486C23C2×D4×D8311yescharacteristic
487C23C2×Q8×D8312yes—
488C23#41×D8312yes—
489C23#1458183712yes—
490C23#1458183711yescharacteristic
491C23#1458183811yescharacteristic
492C23#1458183911yescharacteristic
493C23#1458183912yes—
494C23#1458184011yescharacteristic
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—3321no—
526C2×C4—3321no—
527C2×C4—3321no—
528C2×C4—3321no—
529C2×C4—3322no—
530C2×C4—3322no—
531C2×C4—3321no—
532C2×C4—3322no—
533C2×C4—3321no—
534C2×C4—3322no—
535C2×C4—3322no—
536C2×C4—3322no—
537C2×C4—3322no—
538C2×C4—3322no—
539C2×C4—3322no—
540C2×C4—3322no—
541C2×C4—3322no—
542C2×C4—3322no—
543C2×C4—834no—
544C2×C4—838no—
545C2×C4—838no—
546C2×C4—834no—
547C2×C4—834no—
548C2×C4—834no—
549C2×C4—1664no—
550C2×C4—1664no—
551C2×C4—1662no—
552C2×C4—1662no—
553C2×C4—1664no—
554C2×C4—1662no—
555C2×C4—1662no—
556C2×C4—1664no—
557C2×C4—1662no—
558C2×C4—1662no—
559C2×C4—1662no—
560C2×C4—1662no—
561C2×C4—1662no—
562C2×C4—1662no—
563C2×C4C22⋊C4×D8314yes—
564C22C2×C22⋊C4×D8311yescharacteristic
565C22C4×Q8×D8312yes—
566C22#1912818012yes—
567C22#1912818111yescharacteristic
568C22#1912818211yescharacteristic
569C22C22⋊Q8×D8312yes—
570C22#124×D8311yescharacteristic
571C22C4⋊Q8×D8311yescharacteristic
572C22#1912826611yescharacteristic
573C22#1912826711yescharacteristic
574C22#1912827311yescharacteristic
575C22#1912827411yescharacteristic
576C22#1912827812yes—
577C22#1912828011yescharacteristic
578C22#1912828111yescharacteristic
579C22#1912830112yes—
580C22#1912830312yes—
581C22#1912830812yes—
582C22#1912831012yes—
583C22#1912832812yes—
584C22#1912833012yes—
585C22#1912833012yes—
586C22#1912833112yes—
587C4—24no—
588C4—44no—
589C4—42no—
590C4—42no—
591C4—82no—
592C4—3324no—
593C4—3322no—
594C4—3322no—
595C4—838no—
596C4—6642no—
597C4—1664no—
598C2#391×D8311yescharacteristic
599C2#2491005011yescharacteristic
600C2#2491005211yescharacteristic
601C2#2491019912yes—
602C2#2491020112yes—
603C2#2491329212yes—
604C2#2491329512yes—
605C2#2491340511yescharacteristic
606C2#2491341011yescharacteristic
607C2#2491358111yescharacteristic
608C2#2491358611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×#1441C2
socle, prime radical, cosocleC24×C83C24
hypercenter#1441D83
hypocenterC83#14283
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ #1441

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
14. C2×#14581782C4semidirect
316. C83#14283semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#145817822
C2×#191275702
C2×#191275462
#249057172
#249057162
#249057152
#249057142
#249057132
Dic83⋊#952
C83×#14412
#1428383